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Expansion of (a ± b ± c)²

If x = 2 + ...

Question

If $x = 2 + \sqrt{5}$ then find the value of $x^{2} + \frac{1}{x^{2}}$ .

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Solution

$\begin{matrix} x = 2 + \sqrt{5} x^{2} = {(2 + \sqrt{5})}^{2} = 2^{2} + {(\sqrt{5})}^{2} + 2 (2) (\sqrt{5}) = 4 + 5 + 5 \sqrt{5} = 9 + 5 \sqrt{5} x^{2} + \frac{1}{x^{2}} = 9 + 5 \sqrt{5} + \frac{1}{9 + 5 \sqrt{5}} = \frac{{(9 + 5 \sqrt{5})}^{2} + 1}{9 + 5 \sqrt{5}} = \frac{81 + 125 + 90 \sqrt{5} + 1}{9 + 5 \sqrt{5}} = \frac{207 + 90 \sqrt{5}}{9 + 5 \sqrt{5}} \end{matrix}$

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